A set is a set. A magma is a set with a binary operator. A semigroup is a magma with an associative binary operator. A monoid has a two-sided identity. And a group has two-sided inverses.
I am wondering about one-sided verses two-sided. Under what conditions is an identity element necessarily two-sided? Under what conditions is an inverse necessarily two-sided? And what are the simplest proofs for these?
Theorems I have so far:
A magma may have multiple distinct left-identities or multiple distinct right-identities, but can never have a distinct left and right identity. [1: $\forall x. lx=x$. 2: $\forall x. xr=x$. 1 implies that $lr=r$ while 2 implies that $lr=l$. So either $l=r$ or at least one of 1 or 2 is false.]
Associativity plus the existence of a two-sided inverse is enough to imply that any inverse is two-sided. [If $y$ is the left-inverse of $x$ then $xyx = x(yx)=xi=x$. By associativity, $xyx=(xy)x=x$, which implies that $xy=i$. In other words, $y$ is also the right-inverse of $x$.]
I have a feeling that an associative magma cannot have one-sided identities - but I cannot prove this.